Question

In the given figure, A, B, C and D are points on the circle with centre O. Given, ∠ABC = 62°, find:

1. ∠ADC 
2. ∠CAB

Answer 1

Given:

A, B, C and D are points on the circle with centre O, and ∠ABC = 62°.

From the figure, O lies on BA, so BA is a diameter.

Find ∠ADC:

The angle ∠ABC has its arms BA and BC meeting the circle at A and C.
So it subtends chord AC.

The angle ∠ADC has its arms DA and DC, which also meet the circle at A and C.
So it also subtends chord AC.

Angles standing on the same chord (in the same segment of a circle) are equal.

∠ADC = ∠ABC = 62°.

Find ∠CAB:

Since BA is a diameter, ∠ACB is an angle in a semicircle; hence, ∠ACB = 90°.

In triangle ABC, the sum of angles is 180°:

∠ABC + ∠ACB + ∠CAB = 180°

62° + 90° + ∠CAB = 180°

∠CAB = 180° − (62° + 90°)

∠CAB = 180° − 152°

= 28°.